Uniquely identifying topological order based on boundary-bulk duality and anyon condensation

ABSTRACT Topological order is a new quantum phase that is beyond Landau’s symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R and F matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R and F matrices be experimentally measured? We show that the braidings, i.e. the R matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F matrices can also be measured by comparing the quantum states involving the fusion of three anyons in two different orders. Thus we provide a model-independent experimental protocol to uniquely identify topological order. By using quantum simulations based on a toric code model with boundaries encoded in three- and four-qubit systems and state-of-the-art technology, we obtain the first experimental measurement of R and F matrices by means of an NMR quantum computer at room temperature.

mathematically [1]. We label the anyons in a finite set C by a, b, c, . . . , and the vacuum sector by 1.
There are several major concepts in anyon models which we elaborate in formal mathematics in this section. The first is fusion between two anyons, where two anyons are combined together, or fused, to give an anyon. It is formulated by a fusion rule: where R c ab is the so called R-matrix. If a or b is an Abelian anyon so that there exists a unique fusion channel c = a ⊗ b, the matrix R c ab is reduced to a number R ab . In particular, in an Abelian anyon model, as treated in this work, all the R-matrices can be organized into a single matrix (R ab ) a,b∈C , which we also refer to as R-matrix by slightly abusing the terminology.
The two different ways of fusing three anyons a, b and c are related by the so called F -matrices: In an Abelian anyon model, we have d = a⊗b⊗c and the matrix F d abc is also reduced to a number F abc . In the toric code model, it is easy to see that F abc ≡ 1. We measure this value in the paper using scattering circuit with one additional ancilla control qubit to show that F -matrices are measurable in principle. However, for general anyon models, F -matrices can be nontrivial. For examples, the F -matrix of semions s is F s sss = −1 [2]. The θ a is called topological spin, relating to the ordinary angular momentum spin s a by θ a = e i2πsa and can be determined by Rand F -matrices through the equation Indeed, the right-hand side of the equation involves a little bit of information about F -matrices in a subtle way. In an Abelian anyon model with trivial F -matrices, the equation is reduced to a rather simple one θ a = R aa * .
The effect of twisting (topological spin) is encoded in the modular T -matrix through the relation The fractional spins of anyons can be interpreted as their ribbon structure. Expressing anyons with ribbons implies the relation between twisting and braiding (R k ) c ab = e kπisa e kπis b e −kπisc I where I is the identity matrix of rank N c ab . In the case of k = 2, it gives the effect of double-braiding (moving a around b or moving b around a in a full-circle), where (R 2 ) c ab = R c ba R c ab = e 2πisa e 2πis b e −2πisc I = θaθ b θc I. The data of double-braiding is encoded in the modular S matrix where D is the total quantum dimension defined by D = a d 2 a . The S-matrix is determined by Rmatrices through the relation The fusion and braiding are related by the Verlinde formula The above-mentioned relations (7) and (9) indicate that the modular matrices Sand T -matrices can be deduced from Rand F -matrices. Since Rand F -matrices together can uniquely determine the topological order, we call them the unique identifier of topological order and provide a protocol to measure them.

II. ANYON CONDENSATION AND HALF BRAIDINGS ON GAPPED BOUNDARY
In this section, we elaborate on the mechanism of anyon condensation and half braidings in the context of the toric code model. Toric code model is a special case of quantum double model with group Z 2 (D(Z 2 )). Mathematically, the anyons in QDM with group G can be labeled by pairs (C, π), where C is the conjugacy classes of G and π is the irreducible representations of the centralizers of C. The excitations on the boundary have a topological order given by a unitary fusion category (UFC). This fusion category is the representation category of a quasi-Hopf algebra and is Morita equivalent to the representation category Rep(G). The elementary excitations in the bulk are simple objects in the unitary modular tensor category (UMTC) Z(Rep(G)), the Drinfeld center of Rep(G), which means the bulk is given by the boundary by taking Drinfeld center [3][4][5]. Suppose the boundary is described by a UFC C. Then the bulk, the quantum double or Drinfeld center Z(C) has objects labeled by pairs (x, e x ), where x ∈ C and e x is a half braiding.
The tensor product of the objects in the bulk is given by (x, e x ) ⊗ (y, e y ) = (xy, e xy ), and the braiding is given by c (x,ex),(y,ey) = e x (y). This mathematical structure sheds light on the characterization of a topological order through its boundary.
In our work, we experimentally measured the half braidings on the gapped boundary of toric code model.
Further, the half braidings define the braidings (R matrices) in the bulk and other bulk properties (S and T matrices) can then be deduced. When an m anyon approaches a smooth (white) boundary, it condenses to vacuum, while ε condenses to e.
The bulk to boundary map is 1, m → 1, e, ε → e and Type 1 boundary is also known as {1, e} boundary according to its boundary excitations. This condensation procedure indicates that the m anyon can be created by a local operator acting on the boundary. But when m is moved into the bulk and become a bulk anyon, it was automatically endowed with additional structures called half braidings. The half braidings can be measured by moving this m around the boundary excitations 1 or e along a semi-circle near the boundary, which leads to a trivial or a −1 phase difference. This can be checked by applying all the vertex operators From these half braidings on the Type 1 boundary, we can then define braidings in the bulk and derive the R-matrix. The derivation of all the nontrivial braidings are listed in the following Table S i.
If we exam the Type 2, i.e. {1, m} boundary, we can get another R-matrix (denoted by R ′ ). Its nontrivial   Table S ii.
The following two R-matrices are defined by the half braidings on the two gapped boundaries of toric code lattice: The basis of the above matrices are {1, e, m, ε}. These two R-matrices describe all the braidings in the bulk. Note that R and R ′ are equivalent in the sense of topological consistency equations and are related to each other by taking transpose (this is more clear in the second example D(Z 3 )). Each double-braiding (denoted as R 2 ) can be obtained by combining two braidings together, as illustrated in Table S iii, where two components of R are combined to get one element of R 2 .
With the equations θ a = R aa * , T ab = θ a δ ab and S ab = 1 D R ba R ab discussed in the previous chapter, we can recover all the information about the bulk of the toric code model. For instance, we derive the Sand T -matrices: We can see different version of R-matrices (from different boundaries) give the same Sand T -matrices.
From the above results, we come to the conclusion that the bulk anyons are the boundary excitations equipped with half braidings, which is physically measurable. Braidings in the bulk can be defined by the half braidings on one gapped boundary.
Our scheme can be straightforwardly applied to other Abelian anyon models. We take D(Z 3 ) as an illustrative example which also has trivial F -matrices. The method of deducing the bulk anyons and boundary excitations for quantum double models, in general, is discussed in [4,6]. code lattice on the torus where there is a boundary excitation e 2 . The half braiding between e 1 and e 2 can be measured by first creating an anyon pair e 1 -ē 1 on the boundary, then moving e 1 into the bulk along a semicircle around the boundary excitation e 2 and finally annihilating the anyon pair e 1 -ē 1 on the boundary.
The string operator of the full circle commutes with the string operator connecting e 2 -ē 2 pair because they have no sharing qubit. Thus the full circle string is contractable and this circle operation has no effect on the quantum state. We can therefore conclude that the half braidings between e-class anyons equal to 1.
Non-Abelian anyon models are more complicated and a general measurement protocol is proposed in the next section.
In the language of R-matrices, R a i ,a j = 1 for i, j = 1, 2, ..., m (as illustrated in Fig. S3a). With all these measured R-matrices and the fact that anyons in bulk can be expressed as (a i ⊗ b k ) (i = 1, 2, ..., m; k = 1, 2, ..., n), all R-matrices for bulk anyons can be obtained via, The measurement scheme proposed in this section can be further generalized to QDM for a non-Abelian group G or even to more general topological orders. However, in those more general situations, the braidings between two anyons, in general, are non-Abelian (i.e. not a phase any more). This difficulty can be overcome by performing the measurement of the half-braidings in a family of carefully prepared spaces of ground states with different sets of bulk anyons and boundary excitations. We leave the details and experimental implements to the future.

IV. THE SUBTLETY OF R-MATRIX MEASUREMENT
When preparing our manuscript, we noticed a recent theoretical microscopic construction of anyon data (R-matrices and F -matrices) [11], in which the authors used movement operators and splitting operators to construct the F -matrices and R-matrices.
However, we want to point out the subtlety of measuring the R-matrices in a microscopic lattice model as follows. Fig. S4 shows the graphic representation of the defining process of the R-matrices. In order to get states with two anyons a and b after splitting, the state before splitting can be (ab) and (ba) respectively. We elaborate this on their calculation of the claimed R-matrix of toric code model. In [11], the authors claimed that they did the following calculation involving splitting and moving operation of anyons shown in Fig. S5 to get R m,e = ⟨2|1⟩ = −1. We can see that from |em 1 ⟩ to |1⟩, e is effectively move around m counter-clockwise along a semi-circle, by comparing their relative positions in this process. From |em 1 ⟩ to |2⟩, e is effectively move around m clockwise along a semi-circle. Therefore, ⟨2|1⟩ is the phase factor when e is moved around m counter-clockwise along a full-circle, which is double-braiding S m,e instead of braiding R m,e .

|1⟩ |2⟩
From the above discussion, we can see the subtlety and difficulty in the calculation or measurement of the R-matrices. In fact, to calculate or measure the R-matrices, one has to overcome an obstacle that when moving a around b along a semi-circle (so-called braiding), the space configuration of the final state is different from that of the initial states. For examples, from intial state |em 1 ⟩ to final state |1⟩ is a braiding characterized by R −1 e,m = R e,m ; From intial state |em 1 ⟩ to final state |2⟩ is a braiding characterized by R m,e . We are able to overcome this difficulty by making use of bulk-boundary correspondence and anyon condensation for lattice models with small system sizes.
Furthermore, in Ref. [11], different versions of R-matrices and F -matrices are said to be due to different gauge choices. But in our paper, different versions of R-matrices result from choices of different boundary conditions. There are physical meanings for different versions of anyon data. In addition, we have demonstrated a key principle of topological order, i.e. the bulk-boundary correspondence.

A. Quantum Processor
In our experiment, we perform the half braiding measurement and F -matrices measurement in the 3 and 4-qubit plaquette using a nuclear magnetic resonance (NMR) quantum simulator. All experiments are carried out on a Bruker Ascend NMR 600 MHz spectrometer at room temperature. The processor is a sample of 13 C-labeled trans-crotonic acid molecules dissolved in d 6 -acetone. Their molecular structure and relevant parameters are shown in Fig. 3(f) of the main text. The sample consists of four 13 C 's labeled as C 1 to C 4 and they work as four qubits. The internal Hamiltonian of the system [12] is written as where σ z j is the Pauli-Z operator of the j-th qubit, ν j and J jk are the chemical shifts and the J-coupling strengths between different qubits [13,14], respectively.

B. Half Braiding Measurement: Interference and Tomography Approach
The half braiding of m, e anyons in toric code can be demonstrated in a plaquette surrounded by smooth boundaries consisting of three or four qubits (Fig. 3(a)(b) in the main text) with the following Hamiltonian, They give the ground states A σ z rotation of qubit 3 in the main text Fig. 3(a)(b) leads to a pair of e anyons in the vertexes near Q 3 .
We first prepare the superposition |φ e ⟩ + |φ e ⟩ as initial state: The half braiding is achieved by moving a m anyon along Path 1, which leads to a relative phase: Another trivial Path 2 is taken for comparison. Moving a m anyon along Path 2 has no effect on the initial state: The final states after going through half braiding and trivial path can be differentiated through quantum state tomography. The experimental details and results is discussed in the following.

Experimental Process and Results
Our first experiment is divided into four steps: initialization of the processor, preparation of the superposition state, implementation of the half braiding process and measurement of the final state. We describe each part in detail in the following and the processes of the rest two experiments are similar. 1) Initialization. -As ensemble quantum computing, NMR qubits work at room temperature and the state of the NMR processor at thermal equilibrium is highly mixed [13]. So we need some "magic" to turn it into a pure state (say |00...0⟩), which can serve as the initial state for quantum computing. The thermal equilibrium state of the 4-qubit NMR processor takes the form where ϵ = ℏω 0 /k B T ≈ 10 −5 indicates the ratio of polarized spins and I is a 16 × 16 identity matrix.
Note that the identity does not evolve under unitary propagator, and the only part that contributes to the NMR signal is the remaining term, which is called the deviation density matrix. Hence, we can simplify the analysis by focusing only on ρ dev ≈ 4 i=1 σ z i . Although ρ dev is traceless and unnormalized, it indeed provides the same dynamics as well as much convenience compared with ρ eq when studying NMR quantum computing.
Meanwhile, a pure state |0000⟩ can be expanded in the Pauli basis as This expansion tells that ρ 00 involves multiple orders of coherence except for the identity. The idea to create ρ 00 from ρ dev is to generate high-order coherence step by step while keeping their respective coefficients as required by Eq. (25). This process inevitably involves non-unitary operations due to the change of purity trace(ρ 2 ) from ρ dev to ρ 00 .
The technique to prepare ρ 00 in our experiment is called the spatial average technique [15]. The pulse sequences involved in this process can be seen on the left side of the first dashed lines in the circuits of Fig. S6. Other than regular single-qubit rotations, the initialization involves some U ( 1 2J ) and Gz operators. U ( 1 2J ) operation is used to increase the coherence order by one. For example, U ( 1 2J 12 ) between Q 1 and Q 2 means a unitary evolution under the coupling term π 2 J 12 σ z 1 σ z 2 (see Eq. (17)) with duration t = 1 2J 12 . It is easy to verify that σ x 1 will be transformed to σ y 1 σ z 2 by the evolution U ( 1 2J 12 ) = e −i π 4 σ z 1 σ z 2 , which is how the coherence order increases. Gz operator is the z-gradient field. It is a non-unitary operation to crush all terms in the transverse x-y plane and leave only the longitudinal term. Ideally, one can check that the sequence in Above is the "magic" in NMR quantum computing to initialize the processor. Since the created ρ 00 is not genuinely pure due to the large omitted identity term, it is usually called a pseudo-pure state. In our experiment, the entire procedure of the pseudo-pure state creation takes around 15 ms with each operator realized by optimized pulses and refocusing schemes. We numerically simulated the fidelity and found it to be over 99.8%. In practice, we performed full state tomography after the initialization step and obtained fidelity beyond 99.8%.
and |ψ (4) in ⟩ = φ These two circuits perform the nontrivial half braiding path (Path 1 in Fig. 3(a)(b) of the main text).
2) Preparation of the superposition state and Implementation of the half braiding process. -After initializing the processor, we can perform a standard quantum gate following the circuit diagram in Fig. S7.
To reduce the gate errors in the experiment, all of the control pulses are obtained by the gradient ascent pulse engineering (GRAPE) method and all the pulses have theoretical fidelities over 99.9%. First, the initial superposition state |ψ in ⟩ = |φ g ⟩ + |φ e ⟩ is prepared using a series of Hadamard and CNOT gates. They are realized by a single-qubit rotation e −i π 4 σ y and by exploiting the coupling between qubits, respectively.
Considering CNOT gate between qubit 1 and 2 for the example, it can be realized by modifying the evolution operator U J (t) = e −i π 2 J 12 σ z 1 σ z 2 t from the Hamiltonian (17) as where Z 1 , Z 2 , X 2 and Y 2 stands for e −i π and e −i π 4 σ y 2 , respectively. Because of the varied coupling between different carbon nuclei, we optimize the circuit by taking carbon nuclei C 1 , C 2 , C 3 and C 4 as Q 4 , Q 3 , Q 2 and Q 1 in Fig. S7 respectively such that the CNOT gate can be implemented between carbon nuclei pairs (C 2 -C 3 , C 3 -C 4 ) where the coupling is the largest.
Then we measure the prepared superposition states |ψ in ⟩ for 3-qubit and 4-qubit experiments by full state tomography, obtaining the fidelity over 99% in numerical simulation and beyond 95.46% and 94.50% for 3 and 4-qubit experiment, respectively.
As discussed in the previous section, a half braiding (Path 1 in Fig. 3(a)(b)) will lead to a −1 phase factor for the excited state |φ e ⟩. It becomes a relative phase if we begin with the superposition |φ g ⟩ + |φ e ⟩.
The above process is the quantum state tomography(QST) [16]. In the NMR quantum simulation, QST is the most commonly used method to read out the signal. Here X = e −i π 4 σ x , Y = e −i π 4 σ y and I is a 2 × 2 identity matrix. We can get all information about the final state with QST. Hence, we can estimate the quality of the experimental implementations by computing the fidelity between the theoretical final state ρ th and the density matrix we get from the experiment.
We reconstruct the initial superposition state ρ in and the final state ρ f in after the half braiding and trivial braiding through Path 1 and Path 2 using a maximum likelihood approach [17] and the result is shown in Fig. 3 of the main text. We also calculate the fidelity between the numerical result and the experimental one.
The average fidelities for Path 1 and Path 2 final states are 96.37% and 96.67% for the 3-qubit experiment and are 95.23% and 95.21% for the 4-qubit experiment, respectively.

C. General Half Braiding Measurement
In general, the effect of half braiding can be measured by the scattering circuit with one additional ancilla control qubit as proposed in [18], which is shown in Fig. S8(a). In this circuit, the state before half braiding is prepared as the initial state |φ i ⟩, in general, this circuit can be achieved via an adiabatic approach to prepare ground state (for a toric code Hamiltonian) following certain single-qubit gate on a quantum simulator. Then the half braiding is performed as a controlled operation. Finally, the expectation values ⟨σ z ⟩ and ⟨σ y ⟩ of the ancilla qubit is measured. One can get the following results [18], ⟨σ z ⟩ = Re[T r(Ĥ f |φ i ⟩ ⟨φ i |)] = Re(⟨φ i |Ĥ f |φ i ⟩), ⟨σ y ⟩ = Im[T r(Ĥ f |φ i ⟩ ⟨φ i |)] = Im(⟨φ i |Ĥ f |φ i ⟩).

(28)
For Abelian anyon model, the half braiding (Ĥ f ) leads to a phase factor, and can be obtained from the two expectation values ⟨σ z ⟩ and ⟨σ y ⟩ of the ancilla qubit.
In our experiment, we first prepare Q1-Q3 in Fig. S8(b) to the ground state of the 3-qubit toric code model |ψ 0 ⟩ = |000⟩ + |011⟩ + |101⟩ + |110⟩ with fidelity 95.32%. Then a σ z is applied to create two e anyons on the boundary. This excited state serves as the initial state in the scattering circuit. The half braiding is applied under the control of the ancilla qubit Q4 sandwiched by two Hadamard gates. ⟨σ z 4 ⟩ and ⟨σ y 4 ⟩ measures the real and imaginary parts of the overlap of the states before and after half braiding. We obtain ⟨σ z 4 ⟩ = −0.930 ± 0.004 and ⟨σ y 4 ⟩ = −0.081 ± 0.003 in the experiment. Finally, we normalize these two values such that their square sum to 1, calculate arctan ⟨σ y 4 ⟩ ⟨σ z 4 ⟩ and obtain the phase angle (1.027 ± 0.001)π, which is very close to the theoretical value π. Therefore, we confirm that the m-e half braiding on the smooth boundary leads to a −1 phase factor and then R m,e = −1.

D. F-matrix measurement
As discussed in Section 1, different ways of fusing several anyons to get the same outcome (denoted by fusion trees) are different bases in the fusion Hilbert space. Changing between different bases is done by F move. Fig. 4(b) in the main text shows the two fusion trees for three toric code anyons together with their circle notations connected by F m eem , where ellipses enclose anyons. These ellipses are marked by the fusion outcome of the anyons enclosed.
For Abelian anyons, the fusion Hilbert space is of dimension 1, all the fusion trees are equivalent, and all elements of F are 1. The different trees, as shown in the case of Fig. 4(b) in the main text, are just different perspectives to interpret the same anyon configuration (e, e, m).
We verify this equation in our third experiment using a three-qubit plaquette with smooth boundary. The Hamiltonian and ground state is discussed in the previous section. As shown in Fig. 4(a) of the main text, in Path 1, we first create two e anyons by using σ z 3 , and an m anyon in the plaquette by applying σ x 3 . This corresponds to a circle notation and fusion diagram in Fig. 4(b)(i). In Path 2, we apply an operator σ z 1 to creat two e and then create an m anyon in the plaquette. Suppose the e anyon on the apex angle is far away from m and another e at the left base angle (this can be done by creating two e anyons by a long string operator in a larger toric code lattice), then the m anyon is simultaneously fused with the e anyon at the left base angle once it is created. Then we apply another string operator σ z